The purpose of photometry and colorimetry is to measure quantitatively the radiation and the derived quantities that determine what is seen by a human observer, a camera, or some other image recording device. The goal of this chapter is to lay a foundation so that the reader will understand what is required to record an image accurately, that is, so that accurate information can be obtained from the recorded data. While the accurate information is often used to present a display of the image, it may also be used to derive information about objects in the scene that is used for computer vision or target tracking. Of course, the accurate display of an image is impossible without accurate data. However, to produce an accurate display requires much more. It requires accurate information about the display device and the intent of the observer, who will be judging the image.

Because densitometry is closely associated with quantitative imaging, it is discussed in this chapter. The fundamental difference of densitometry from photometry and colorimetry is that densitometry is concerned with the measurement of physical quantities of display media. The density of a colorant on paper or dyes on film can be related to the appearance of the image that is produced using these means. The relationship is usually approximate. However, since it is so commonly used in the printing and film industries, it is appropriate to discuss its uses here.

The previous chapters have discussed the various tools and properties that are used to characterize a digital image. The image begins as a reflective or radiant source; a distribution of energy is sensed by a device in one or several bands; the sensed signal is converted from analog to digital format and stored using a finite number of bits. The characteristics of an image include properties that are inherent to the content of the image, to the source of the energy distribution and to the sampling and quantization of the image.

The characteristics can be deterministic, such as the size of the image, or stochastic, as in the case of the signal-to-noise ratio in a recorded image. In this chapter, we will consider the various characterizations and their effect on the processing of the image. The characteristics of an image or class of images are important to determine the optimal recording, display, coding and processing. In general, if we are concerned with a specific image, the characteristics are deterministic, since they can be determined by measurements on the image. Such parameters as the mean, minimum and maximum values are examples. If we are concerned with a class of images, it is reasonable to treat the class as a statistical ensemble and characterize it by statistical parameters. Mean, minimum and maximum are examples, but in this case the values are interpreted differently.

Introduction

A look-up table (LUT) is basically a function from one space to another that is defined in terms of a few samples, their corresponding function values, and a method to calculate any particular mapping from those samples. Mathematically, the LUT is defined as L[{(xk, f (xk)}, I(x)], where {xk} are the samples in the domain space, {f (xk)} are the corresponding function values in the range space, and I(x) is the function, or algorithm, that is used to compute the value in the range space for an arbitrary point in the domain space, x. The function I(x) interpolates the output if the point x is within the convex hull of the sample set {xk }, and extrapolates the output if it is not.

Look-up tables are a simple and computationally efficient way to generate nonlinear and nonparametric functions. Because of their efficiency and ease of implementation, look-up tables are often used to compute standard functions, such as sinusoids and exponentials. The accuracy of the tabularized function depends upon the resolution of the table. The key to the efficiency is that the interpolation between elements in the table is simple and fast. This means that accuracy depends on the resolution of the table, rather than the approximation of the interpolation to an ideal functional form.

When an analog signal is transformed to a digital one, it is necessary to limit the representation to a fixed number of bits. This chapter discusses the best way to distribute those bits across the range of possible values. The quantization problem is no different in two dimensions than in one dimension. The first task of quantization is the definition of the term “best.” This task requires not only the definition of an error metric, e.g., mean square error, but also the definition of the space in which the error is measured. The choice of the appropriate space is determined by the physical properties of the image that is being digitized, as well as the intent of the user of the digital data.

Most mathematically based quantization schemes use the mean square error as the error metric because of the ease of analytical manipulation. Since the eye is not a mean square error detector, quantizing for minimum mean square error (MMSE) is not usually visually optimal. However, such quantization rarely results in unacceptable images if an appropriate space is chosen for the quantization. Subjective quantization schemes are common and are often implemented using a look-up table (LUT).

We will first briefly consider the selection of an appropriate space for the image data. After a space is chosen, mathematically optimum quantization can be obtained by use of the MMSE criterion. We will show how to use this for both monochrome and color images.

Digital imaging is now so commonplace that we tend to forget how complicated and exacting the process of recording and displaying a digital image is. Of course, the process is not very complicated for the average consumer, who takes pictures with a digital camera or video recorder, then views them on a computer monitor or television. It is very convenient now to obtain prints of the digital pictures at local stores or make your own with a desktop printer. Digital imaging technology can be compared to automotive technology. Most drivers do not understand the details of designing and manufacturing an automobile. They do appreciate the qualities of a good design. They understand the compromises that must be made among cost, reliability, performance, efficiency and aesthetics. This book is written for the designers of imaging systems to help them understand concepts that are needed to design and implement imaging systems that are tailored for the varying requirements of diverse technical and consumer worlds. Let us begin with a bird's eye view of the digital imaging process.

Digital imaging: overview

A digital image can be generated in many ways. The most common methods use a digital camera, video recorder or image scanner. However, digital images are also generated by image processing algorithms, by analysis of data that yields two-dimensional discrete functions and by computer graphics and animation. In most cases, the images are to be viewed and analyzed by human beings.

The primary digital image recording devices are the digital scanner and the digital still camera. In recent years, these devices have become commonplace, leading to a proliferation of digital images. In this chapter, we discuss these devices in detail, focusing on their limitations and variations. In addition, we will discuss hyperspectral imaging, which makes use of more than three bands to obtain information that can be used in multiple illuminant color reproduction and image segmentation and classification.

Scanners

To process images from scanners and digital cameras effectively, it is necessary to understand the transformations that affect these images during recording. There are several approaches to sampling the two-dimensional image. The first scanners used a single sensor and moved the medium in both orthogonal directions. With the advent of the CCD array, it was possible to project a scan line onto the sensor and move the medium (or sensing array) in the direction orthogonal to the linear array. Finally, it is possible to use a 2-D sensor array to sample the medium without movement during the scan. The most common desktop scanners record the image data using a row of sensors. Thus, we will concentrate on that technology.

The “paper moving” designs consist of both sheet-fed and flat-bed scanners. The “sensor moving” designs include hand-held scanners, which require the user to move the instrument across the paper, as well as flat-bed scanners. The primary sensor types are charge-coupled device (CCD) arrays and contact image sensor (CIS) arrays.

For any type of structured analysis on imaging systems, we must first have some model for the system. Most often it is a mathematical model that is most useful. Even if the model is simplified to the point that it ignores many real world physical effects, it is still very useful to give first order approximations to the behavior of the system.

In this chapter, we will review the tools needed to construct the mathematical models that are most often used in image analysis. It is assumed that the reader is already familiar with the basics of one-dimensional signals and systems. Thus, while a quick review is given for the various basic concepts, it is the relation between one and two dimensions and the two-dimensional operations that will be discussed in more depth here.

As mentioned, the mathematical models are often better or worse approximations to the real systems that are realized with optics, electronics and chemicals. As we review the concepts that are needed, we will point out where the most common problems are found and how close some of the assumptions are to reality.

Images as functions

Since images are the main topic of this book, it makes sense to discuss their representation first. Images are, by definition, defined in two spatial dimensions. Many, if not most, images represent the projection of some three-dimensional object or scene onto a two dimensional plane.

Images can be considered representative of random processes. Given the values of pixels in a well defined region, it is not possible to predict exactly the values of pixels outside of that region. For most images, it is likely that there is some relation between the pixel values inside and outside the known region, but the relationship is statistical. The fact that most images are the result of the measurement of radiation means that there is always some uncertainty about the exact value of a pixel, even within any known region. In fact, deterministic images are usually of little interest. From an information theoretic viewpoint, it is the uncertainty of the values of pixels that makes the information conveyed by the pixels important. This appendix will review the fundamentals that are assumed as a prerequisite for treating the various aspects of noise and stochastic models that are used in this text.

This appendix gives only brief definitions of the terms that we will use repeatedly in the text. It will only briefly discuss the elementary properties and concepts of stochastic processes that are necessary for the description of many of the imaging modeling processes discussed in the main text. Thus, this should be seen as a refresher of material that the reader has seen previously, or perhaps it indicates the material that the reader will need to learn from some more appropriate text in order to understand certain parts of this text.

This book is entirely devoted to the area of visibility algorithms in computational geometry and covers basic algorithms for visibility problems in two dimensions. It is intended primarily for graduate students and researchers in the field of computational geometry. It will also be useful as a reference/text for researchers working in algorithms, robotics, graphics and geometric graph theory.

The area of visibility algorithms started as a sub-area of computational geometry in the late 1970s. Many researchers have contributed significantly to this area in the last three decades and helped this area to mature considerably. The time has come to document the important algorithms in this area in a text book. Although some of the existing books in computational geometry have covered a few visibility algorithms, this book provides detailed algorithms for several important visibility problems. Hence, this book should not be viewed as another book on computational geometry but complementary to the existing books.

In some published papers, visibility algorithms are presented first and then the correctness arguments are given, based on geometric properties. While presenting an algorithm in this book, the geometric properties are first established through lemmas and theorems, and then the algorithm is derived from them. My experience indicates that this style of presentation generally helps a reader in getting a better grasp of the fundamentals of the algorithms.

Problems and Results

A simple polygon P is said to be an LR-visibility polygon if there exists two points s and t on the boundary of P such that every point of the clockwise boundary of P from s to t (denoted as L) is visible from some point of the counterclockwise boundary of P from s to t (denoted as R) and vice versa (see Figure 4.1(a)). LR-visibility polygons can be viewed as a generalization of weak visibility polygons (discussed in Chapter 3). We know that a simple polygon P is a weak visibility polygon from a chord if there exists two points s and t on bd(P) such that (i) s and t are mutually visible, and (ii) every point of the clockwise boundary from s to t is visible from some point of the counterclockwise boundary from s to t and vice versa. If the condition (i) is removed, then it defines LR-visibility polygons, which contains weak visibility polygons as a subclass. It can be seen that if a point moves along any path between s and t inside an LR-visibility polygon, it can see the entire polygon. LR-visibility polygons are also called streets and this class of polygons was first considered while studying the problem of walking in a polygon. For more details, see Icking and Klein [200] and Klein [219].

Problems and Results

Assume that a point-robot moves in straight-line paths inside a polygonal region P. Every time it has to change its direction of the path, it stops and rotates until it directs itself to the new direction. In the process, it makes several turns before reaching its destination. If straight line motions are ‘cheap’ but rotations are ‘expensive’, minimizing the number of turns reduces the cost of the motion although it may increase the length of the path. This motivates the study of link paths inside a polygonal region P. For more details on applications of such paths, see the review article of Maheshwari et al. [253].

A link path between two points s and t of a polygon P (with or without holes) is a path inside P that connects s and t by a chain of line segments (called links). A minimum link path between s and t is a link path connecting s and t that has the minimum number of links (see Figure 7.1). Observe that there may be several link paths between s and t with the minimum number of links. The link distance between any two points of P is the number of links in the minimum link path between them.

The problem of computing the minimum link path between any two points inside a simple polygon were first studied by ElGindy [126] and Suri [318].

Problems and Results

The notion of weak visibility of a polygon from a segment was introduced by Avis and Toussaint [42] in the context of the art gallery problem. They considered a variation of the problem when there is only one guard and the guard is permitted to move along an edge of the polygon. They defined visibility from an edge vivi+1 in a simple polygon P in three different ways.

1. P is said to be completely visible from viυi+1 if every point z ∈ P and any point w ∈ vivi+1, w and z are visible (Figure 3.1(a)).

2. P is said to be strongly visible from vivi+1 if there exists a point w ∈ vivi+1 such that for every point z ∈ P, w and z are visible (Figure 3.1(b)).

3. P is said to be weakly visible from vivi+1 if each point z ∈ P, there exists a point w ∈ vivi+1 (depending on z) such that w and z are visible (Figure 3.1(c)).

If P is completely visible from vivi+1, the guard can be positioned at any point w on vivi+1. In other words, P and the visibility polygon V(w) of P from any point w ∈ vivi+1 are same. If P is strongly visible from vivi+1, there exists at least one point w on vivi+1 from which the guard can see the entire P, i.e., P = V(w). Finally, if P is weakly visible from vivi+1, it is necessary for the guard to patrol along vivi+1 in order to see the entire P.

Problems and Results

The visibility graph is a fundamental structure in computational geometry; some early applications of visibility graphs include computing Euclidean shortest paths in the presence of obstacles [249] and in decomposing two-dimensional shapes into clusters [306]. The visibility graph (also called the vertex visibility graph) of a polygon P with or without holes is the undirected graph of the visibility relation on the vertices of P. The visibility graph of P has a node for every vertex of P and an edge for every pair of visible vertices in P. Figure 5.1(b) shows the visibility graph of the polygon in Figure 5.1(a). We sometimes draw the visibility graph directly on the polygon, as shown in Figure 5.1(c). It can be seen that every triangulation of P corresponds to a sub-graph of the visibility graph of P. The visibility graph of a line segment arrangement is defined similarly, where the endpoints of the line segments are represented as the nodes of the visibility graph.

Consider the problem of computing the visibility graph of a polygon P (with or without holes) having a total of n vertices. The visible pairs of vertices in P can be computed by checking intersections of segments connecting pairs of vertices in P with each polygonal edge of P. This naive method takes O(n3) time as in the first algorithm for computing the visibility graph given by Lozano-Perez and Wesley [249].